Understanding the Cosine Function

What is the Cosine Function?

The cosine function, denoted as cos(θ), is one of the fundamental trigonometric functions. It relates the angles of a right triangle to the ratios of the lengths of its sides. In the context of the unit circle, cos(θ) represents the x-coordinate of the point where the angle's terminal side intersects the circle.

Formula and Definition

For a right-angled triangle with an angle θ, the cosine function is defined as:

\[\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\]

Where:

• "adjacent" is the length of the side adjacent to the angle θ
• "hypotenuse" is the length of the longest side of the right triangle

Calculation Steps

1. Identify the angle θ in degrees or radians.
2. If the angle is in degrees, convert it to radians using the formula: θ (in radians) = θ (in degrees) × π/180°.
3. Apply the cosine function: cos(θ).
4. Round the result to the desired number of decimal places.

Example Calculation

Let's calculate cos(60°):

1. θ = 60°
2. Convert to radians: 60° × π/180° = π/3 radians
3. cos(π/3) = 0.5

Visual Representation

This diagram illustrates cos(60°) in the unit circle. The green line represents the cosine value, which is the x-coordinate (0.5) of the point where the angle's terminal side intersects the circle.